Question

Find:

`int ((x - 3))/(x - 1)^3 e^x dx`

Answer 1

`int ((x - 3))/(x - 1)^3 e^x dx`

= `int (x - 1 - 2)/(x - 1)^3 e^x dx`

= `inte^x((x - 1)/(x - 1)^3 -2/(x - 1)^3) dx`

= `int e^x (1/(x - 1)^2 - 2/(x - 1)^3) dx`

It is the form of ∫ e^x [f(x) + f'(x)] dx = e^x f(x) + C

Where f(x) = `1/(x - 1)^2`

∴ f'(x) = `-2/(x - 1)^3`

So, our equation becomes

`int e^x (1/(x - 1)^2 - 2/(x - 1)^3) dx = e^x/(x - 1)^2 + C`